1 SETS, RELATION AND FUNCTION SECTION-I SET THEORY 1.1. Introduction be expressed in basic and In fact all mathematical ideas can unifying idea of mathematics. Set is a in one form or the other. whole of the business mathematics, the set theory is applied terms of sets. In almost 1.2. Def. of a set A set is a collection of well defined and different objects. that we are given a rule with the help of which we can say By the words 'well defined' we mean of objects set or not. The word 'different' implies that repetition whether a particular object belongs to the is not allowed. for the word set when the elements 'collection' are also used as synonyms The words 'family', 'class', are themselves sets. Element of a Set element of the set. Each object of the set is called an Examples of sets (i) The set of integers from 1 to 100000. of a week. The set of days The set of all states of India. The set of even integers. (iv) ii) x*=1. (v) The set of all solutions of equation Set Notations letters A, B, C,. Sets generally denoted by capital are small letters a, b, e.. The elements of sets are denoted mostly by Some Standard Sets 2,3,4,. N Set ofallnatural numbers 1, W Set of all whole numbers 0, 1, 2, 3, 4,.. I or Z Set of all integers numbers Q- Set of all rational R Set of all real numbers 2 SPECTRUM DISCR MAT Methods of Designating a Set FUNCTION A setcan be specified in two ways RELATION AND SETS, Enumeration Method: (1) Tabular, Roster or Examples When we represent a set by listing all its elements within curly brackets{}, A -f1,2,3. odd integer is called the tabular, roaster or enumeration method. separated by. B fr:r is an ) A set of vowels : A {a e, i. o, u}. commas C-irir is a multiple of 6} set or a unit Singleton Set singleton (u) A set ofpositive even integers upto 10: B = {2, 4, 6, 8, 10 element is called a set containing only single x S 40} = {6} ui) Asetof odd natural numbers: C= {1, 3, 5 , . . A perfectsquare and 30 {r:x is a {2} r=4} = = Example:A 2) Selector. Set-builder or Rule Method: positive integer satisfying do not ist all the elements but the set is B=fx:r is a In this method. we represented by specifving thedae For example. cifying the defining propeny C {3 Empty, Null or Void Set: and is denoted by o (read as phi) A r r i s a vowel in English alphabets} element, is called a null set contains no A set which B { I I s a positive even integer up to 10} satisfying x"= C trrisan odd positive integer Examples: A ={r:x isa positiveinteger Here () or () means such that. r=9} B {xx is afraction satisfying Note 1. The order of elements in set is immaterial a Sub-Set, Super-Set called sub-set of B and B is called super-set of A. Thus (2. 5. 9. 11}. {2.9, 5, 11}. {11,5,2, 9} represent the same set. A is a element ofa set B, then A is Ifevery element ofa set is a super set of A. 2. Repetition of elements is not allowed in a set. Or if rE A x¬ B, then A is a sub set of Band B Membership of a Set CB and B 3 A. We writethese as A If an object r is a member of the set A, we WTite x E A, which can A is contained in B or B contains A. be read as 'r Thus A C B means A contains r. Similarly we write x A to show that x is not a member belongs to A or . set A. of the Note 1. Since every element of A belongs A Example. Let A {1.2. 5, 7,9, 10 = ACA every set is sub set of itself. Here 5E A, but 6 £ A. considered to be a Subset of every set 2. The empty set o is Type of Sets elements then number of subsets of A is 2". 3. If set A has n Finite Set Let A {1,2,3, 4, 5, 6, 8, 10; = Example. A set is said to be finite if it has D={2,7, 8, 1} finite number of elements. and B {2,4,6, 10},C={1,2,7,8}, Examples: Now every element ofB is an element of A A {2,4,6,8 BCA B {x:xis astudent of Modi College, Patiala} Again 7E C, but 7 A Set of months of the year C ¢ A ie., C is not a sub-set of A. Set of even natural numbers is member of D. member of C and every member of C a less than 100. Now every member of D is a Infinite Set CCDand D CC A set is said to be infinite if it has an infinite number of elements. also write C E D and D SC. In this case we can sPLCTRUNM DISCRETEM Equality of Sets Two sets A and B are said to be cqual if both have the same elements. In other w AND FUNCTION SETS, RELATION ILL USTRATIVE EXAMPLES B are equal when ever clement of A IS an element of B and every element of B is an words, two sets B. ment :eIf ACB and B C A.then A Example. A1.2,3, 4, 5, 6. 7, 8, 9. 10 form: 2 in Roster C = {x:rF- sets u) B a.ais a natural number and 1 s r s 10 Example 1. Write the following EN:Ix| 4 25} (in B fx Ir E N:? - Here A B A 7 but less than 28 positive multiple of 3 and D= { r : r i s a Proper Sub-set () x =5, - 5 A non-empt set A is said to be a proper subset of B if ACB and A * B. =25 Sol. (0x* But r E N, X5 Note 1.9 and A are called improper subsets of A. Power set A-{5 EN The power set of a finite set is the set of all sub-sets of the (i) x=ras x given set. Power set of A is denss. x = 1, 2, 3, 4 xs4 x S 4 Eample. Take A = {1, 2. 3 yPA B{1,2, 3, 4} x=1,2 PA) o. },(2) (3). {1. 2), {1, 3. 12,3}. (1,2, 3} -3x+2 0 ( r - 1) (x -2) (ii)x Universal Set C ={1,2} 7 If all the sets under consideration are sub-sets of a fixed set U, then U is called universalsset. positive multiple of 3 and (iv) Since xis a a Example. In Plane geometry, the universal set consists of points in a plane. xis a multiple of 21 Comparable and Non-comparable Sets D-{21 builder notation: set Two sets are said to be comparable if one of the two sets is a sub-set of the other. of the each following sets, using Example 2. Redefine (6) {0, 3, - 3, 6, 6. 9. Example. Let A = {2. 3. 5}, B = {2,3, 5, 6}. C= {1, 5} (a)-2,-4,-6,...} Here ACB () {2, 3, 4, 6, 8, 9, 10, 12, 14, 15,.. A and B are comparable sets. On the other hand A ¢ C,c¢A A. C are Non-comparable. (e) {0, 1, 2, . 9 9 , 100} Order of a Finite Set Sol. () Let A ={-2,-4,-6,. =-2x:x EI} {3 xEl} -3, 6, -6, 9....} = x: The number of different element of a finite set A is called the (6) Let A {0, 3, order of A and is denoted by O(A). (c)Let A = {2,3, 4, 6, 8, 9, 10, 12, 14, 15,. Cardinality : Number of different elements in a set is known as its cardinality. Example If A = {2,3, 6, 8}, then O(A)=4 {x:x=2 nor x =3n:nENN} Equivalent Sets (d) Let A Two finite sets A and B are said to be equivalent sets if the total number of elements in A is the total number of elements in B. equal to S.t. 0 S r s 100} (e) Let A ={0, 1, 2, . . 9 9 , 100} = {x:x El Example. Let A = {1,2, 3, 4, 6}, B={1,2,7,9, 12} Example 3. Define geometrically, the following set OA)-5 = O(B) A and Bare equivalent sets. (a)txE R: |x| s 3 } (6)x EZ,: |x| s 3} We write the above fact A B as (){x EN, |x|s 3} () {x. ),x,yE R, r+y=25} SPECTRUM DISCRET MAT 3} = {r ER:-3 srs 3} Let A {x E R: |x| = s Sol. (o) FUNCTION The set A is represented by RELATION AND ETS, ={r: x is real and x <x} b) X Since r *r-x<0 0 x(x- 1)<0 line from P to Q including the points P and Q. by the points on the 0< x<1 (6) Let A={r E Z, : |x| s 3} = {-3, -2, -1,0, 1,2,3 and 0 <x < 1} the points marked by dot on the line PQ. X= {x:x is real The set A is represented by T P 2 X= T 0 32 (c)A = {1,2,3, 4} x<18} 3} {1,2, 3} { x : x is positive integer and c)Let A {* EN, |x| s = B The set A is represented by the points marked by dot on the line PQ {1,2, 3,4} yes, A B ? Justify. 0 } empty P +2 = andr-3x +6 = 0 A = {x:x-5x Is the set Example 5. (d) Let A={(x,y);x,y ER; *+y=25} x-3x+2=0} and {x:*-5x+6-0 The set A is represented by all the points which lie on the circle whose centre is is at Sol. A - ( r - 2 ) r - 3 ) =0 *=2, 3 at the the n Nowx-5x +6=0 point (0, 0) 0 I=1,2 r-1) (r -2) = radius as shown below: Alsox-3 x+2=0 +2 =0 x-5x+ 6 0 and x-3x ris there is a value 2 ofr which satisfies set. A is not an empty A-2} sets: Enumerate elements in following Example 6. (c){rEC| x* +1 =0} (a) frERx-3x+2 =0} (6) {rER|r*+1 =0} = {rER| x*-3x +2 =0} Sol. (a) Let A Solution Set: x-3x +2 =0 x-2x-Ix +2 0 Example 4. Determine which ofthe following pairs of sets are equal: ( - 2 )( - 1)=0 ()S={x x is an integer divisible by both 3 and 2} and Q= {6, 12, 18, 24,.} x= 1,2 (b) X={x:x isreal and r<x} and T -{x:x is real and 0 <x < 1} A {1,2} ) L e t A = {1,2,3,4). B = {r:x is a positive integer and x<18. =0} (6) LetB {rER|** +1 Sol. (a) S= {x:x is an integer divisible by both 3 and 2} +1 =0x=# V-1¢R {x:x=3n or x = 2 n where n E} Set B has no Solution f. 9,-8,-6,-4, -3, -2, 0, 2, 3, 4, 6, 8, 9,. B. Q-{6, 12, 18, 24,...} (c)Let D = {xEC| x +1 =0} Since 6 E S and -6 £Q r+1 0 *=t -1 >xti¬C. SQ D={i, i SPECTRUM DISCRETE MATHEMAT. is 56 more Evample 7. Find the cardimal number of each set: AND FUNCTIOON of subsets of the first set SETS, RELATION number The total ( ) A i : -25,3 r=6} (i) Power set P(B) of B have m andn elements. values of m and n. {1,4, 5, 9) 10. Two finite sets second set. Find the - 5} Example of the (i) A ={x:xEN, () B {6, 7, 8,9,. total number of subsets Then, Sol. () A = {r:r = 25, 3 x =6 than the elements respectively. m and n two sets having and B be Sol. Let A Since a = 25 x=£5 and 3 x =6 *=2 A =2", Number of subsets of A -g B 2". Card (A) =0 ie., # (A) = 0 Number of subsets of 2"-2"= 56 It is given that (i) Here B {1,4, 5, 93 2"(2--1) =2'(2'-1) P B)= 0, {1}. 4}, {5}, {9}, {1.4, {1, 5}, {1, 9}, {4,5}, {4,9), {5, 9). {1,4, 5), {1,4, 9 n=3 and m - n =3 Cardinal number of (P (B)) =16 4,5,9.I,5,9}, {1,4, 5,9 n=3 and m =6. has 2" subsets. n distinct elements () As x = 5 Prove that a set containing Example 11. Sol. Let A = {aj,a2.43 x tv5 a N Cardinal number is distinct. where a, 's are be made in "C, way. 0 S r S n. Hence, () Cardinal number iso elements of the set A can A selection of r objects from the contains r elements. Example 8. List all the members of the power set of each of the following sets : there are "C, subsets of a A which no elements is "Co. (a) A a , b, 2, 3 Number of subsets of A containing (6)C {ta}. {b}} (c) D {d. {o}} A- {a, b, 2, 3} Number of subsets of A containing1 element is "C Sol. Here P(A)= {¢. {a}, {b}, {2}. 13}, {a, b}, {a, 2}, {a, 3), {b, 2}. {b, 3}, {2, 3}, Number of subsets of A containing 2 element is "Cz. {a, b, 2), {a, b, 31 b, 2,3), {a, 2, 3}, {a, b, 2, *************" *** *****"******* b) C= {{a). {b}} 3 ... ********* **** P(C)-{o.tta}}. {tb}}, {{a), {b}} ******* *******************. Number of subsets of A containing n elements is "C (c)D=o. io}} P(D) = {p. {p}. { p}}. {o. {o}}} Hence, the total number of elements of A = "Co+"C, + "C2 +. Example 9. Let A {r. s, t, u, v, = w}, B = {u, v, w, x, y, z}, C= {s, u, y, z}, D = {u, v}, E = {s, u} and "n F= {s}. Let X be an unknown set. 2" Determine which sets A, B, C, D, E or F con equalX if we are given Example 12. What is the number of subsets of a set having n elements. Write down all the proper subsets ( XCA and XCB (ii) X gB and X cC ofthe set {1,2, 33. (ii) X A and X ¢C (iv) XCB and X ¢C. Sol. The number of subsets o f a set havingn elements = 2". Sol. (i) The only set which is a subset of both A and B is D. Notice that C, E, and F are not subsets of B sincesE C, E, F and s B. No. of proper subsets ofa set= 2" -1 (ii) Set X can equal C, E or F since they are subsets of C and these are not subsets of B. Let A {1, 2, 3), n=3 (ii) Only B is not a subset of either A or C. D and A are subsets of A; C, E, F are subsets of C. Thu X= B. No. of proper subsets of A 2* -1 =7 (iv) Both B and D are subsets of B and are not subsets of C. Hence X = B or X = D. Proper subsets of A arep, {1}, (2}. {3}, {1, 2}, {1, 3}, {2, 3}. 11 10 SPECTRUM DISCRETE MATHEMAT FUNCTIONN EXERCISE 1.1 SETS, RELATION AND the cardinal number ofeach set: {6, 7, 8, 9,... least two 1. List the elements in each of the following sets using braces and ellipses where necessarv 12. Find A-x:r E N,?=5) (i) B Use set builder notation for at set of the integer. (a) r:r is a natural number divisible by 5} (6)xx is of the power a negative odd integer) List some of the elements 13. (c){r:x is an even prime number} ()xx-1 is an integer divisible by 4 of these elements. ANSWERS ( ) r : x i s an integer divisible by 2 and by 5} (c) 2) 2. Write the following sets in Roster form: (b) -1,-3, -5,-7, . - 10, 0, 10, 20,...} 5, 10, 15,20,.. ..-20, () (e) x : r is a vowel beforeg in the English alphabet 1. .. {1,2,3, 4, 5, 6, 7, 83 1, 5, 9, 13, (in) 13, 17, 19, 23, 29) (d).7,-3, n {xEN:x is a prime numberbetween 6 and 30} (i) 17, 11, () a, e} (üi) fx E N:3x+5 <31} (iv) -2,-3} (b) (c)ti,- 3. (a) 1, 2} (n') {rr+5x+6=0 integer from 5 to 11} 4. {r:x is positive 3. Enumerate the element , in the following sets : 3, 4, 5} x:r=2n, n = 1, 2, is 9} (i) which the sum of two digits (a) rE R*-3x+2 =0} (b) {x ¬ R/r+1 =0} (c) E Cs digit number in +1 =0} (ii) fr:r is a two 1 sks39} 4. Describe the following sets in set builder form : 5. (a) {2 k+1: kEl suchthat =2n:nEl} (c) {x:r A (5,6,7,8, 9, 10, 11} i) B= {2,4, 6, 8, 10} (6) frEQ: 1<x<1} 8. {2} 9 Yes 7. (i) C {18, 27, 36,45, 54, 63, 72, 81,90} 6. A,C (b) True (c) True (d) True 5. Describe the following sets using set builder notation 10. (a) True (c) True (d)True (6) False 11. (a) True (g) False (o) 13,5, 7,9,..77, 79} (e) True True b) The rational numbers that are strictly between -I and 1 12. 0 (ii) natural number), (c) The even integers {1, 2, 3,4, 5}, E, N, W; E = {r:r is a 13. . 6. Which ofthe following sets are null sets? W {x:x is whole number} A :xEN,x< 1} (i) B={x:x+4-4 (ii) C= {rix<l andx>3} Operation on Sets What is the set fx:r E R,r =9,2x =4}? 1.3. Venn Diagrams Venn illustrated by certain diagrams called Venn diagrams. In a 8. Whatisthe set {r:x E R,r=4,x*-3 x+2=0} ? The relations between sets can be circle within and any sub-set of U is represented by a diagram, universal set U is represented by a rectangle 9. Isthe set A= {x:r=8 and 2 x+3 0} empty ? Justify a rectangle U. 10. Let A= {x:x is rational and 0 srs2} and let 1.4. Complement of a Set B x:xis rational and 1 Srs 3}. Then the Let A be a subset of universal set U. Indicate whether each of the following is true or false. elements of U which complement of A is the set of all those of A by (a) 0-5 E A, (6) 2 EB do not belong to A and we denote complement A or A". (c) 26 E A. or 26 E B () 3 xlr EA and x E B We can write I1. Let A = {0, 2, 3}, B {2,3} and C {1,5, 9}. Determine which of the following statements are Ax:xE U,x £A } true? Give reason for your answer A is Shaded like ZZNIZ The complement of A is the shaded region (a) 3EA (6) {3} EA {2,4,6,8, 10} A= {4,8} then A {2,6, 10} = c13} CA ()BCA ACB Example: IfU = )OEC ) EA Note: U =o and p = U, (A° =A 12 3 1.5. min of Two sets SETS,REATNAND FUNCTKS 1fA ad h be 1wogiven t t theT their tumio f Two Sets is the se 1.7. Difference dhoz diemets o comeetng of all the elemerits of A tether with aff the he se A mdB is e shouid n ciements The difference oftao set We denose this y A - B. repet the B. The ar eierer of two e t s A A which do not belona to we w r e in synbos, insysbs. A U B {z zEA rzEE A-B-z z E And ELmpk L sometimes writte 2s Al B. A 1.2. 3.5. . AUBis Shaded like Z A -B isalso d. A -Bis Shaded lke ZIZ z Example LaA = {a. b. c. e j. B-2.4. 6; The unicm of two sets A and B tc. defzi AUB- 1.2.3. 4, 5, 6, %; ThenA B-a. b; A - B Note B - A z A na iemly of ses, ten ter nion is dencsedby A, A,UAz UA Symmetric Difference of Two Sets UA (A - B) B - A) then the set IfA and B are ay two ets, by AAB 16. Intersection of Two Sets s called syTmmetric of A difierence is denoted nd B and consists difference of AandB In oher o r d s , the symmetric and B sets A The imersection of two ses A and B. denoted ty An B that belong to exactiy one of the of all the elements sthe se of all eiemens, whch are conmon to A and B. and not to both w e wTLe In symbos. In synbols, A B z z E A and r E B; AAB-z: rEA nd: B) or z E Bandr£ A) A ABis Shaded like Z Z I 2 - (AUB)-(ANB) ELampie Let A= {2. 4, 6. 8. 19, 12 {1. 2, 4;. B =1,2. 3. 5. 6; Example. Let A = AOB is Shaded like Dz B 2.3. 5.7. 11 The intersection of two sets A and B AAB = (A B ) U (BA) = ( A - B ) U B - A) - 14 U13. 5, 6 = {3. 4 . 5, 6} A B:2 Algebra of Sets Note: if A. Az A is a finne farnily ofsets, then their intersection is denoted by 1.8. Some Fundamental Laws of Table Basic Laws of Set Theory Name Identity A or A, nA, riA:| AUO-A ldentity Laws Disjoint Sets AnU-A AUU=U Domination Laws IfAand two given sets such that A n B =p. then the sets A and Bare said to be disjoint. An AUA=A E1ample. Let A a, b, c.d. ldempotent Law AA A B-1, m. n. pi. Complementation Law AnB A and B are disjoint sets (A) A Commutative Laws Thus A and B are disjoint sets. AUB BUA 15 14 SPECTRUM DISCRETE MATHEMAT FUNCTION ANB BNA SETS, RELATION AND |AUBUC) =(A U B) UUC I . Commutative Laws two sets, then An(BC)=(A n B) nc Associative Laws Statement. IfA and B are any ANB=BNA (in An (BUC) =(A n B) U(A NC) AUB=BUA () ={x:xEBorx ¬ A} AUBC)=(A U B) (A UC) Distributive Laws Proof.() L.H.S. =A UB EA orx EB} ={r:x AUB} {x:xE (AUB=A' NB xrEB U A }= B U A De Morgan's Laws (A NBy =A' UB R.H.S. AU(ANB)=A (i) L.H.S. =ANB {x:xE B and x E A} and x E B} = Absorption Laws fx:rEANB} = {x:x E A AnAUB) =A {xxE B O A }= B N A AUA U R.H.S. Complement Laws AA =¢ IV. Associative Laws sets, than Statement. IfA, B and C are any three L. ldempotent Laws N B) n c UC (i) An(BnC) =(A (9AU(B UC)=(A B) U Statement. IfA is any set, then Proof. () L.H.S. =AU (BUC) (AUA=A (i)ANA =A E(B U C)} irixE AU(BUC)} {r:rE Aorr = = Proof. ()LH.S. =AUA {x:(«E Aorx E B) orx C} ¬ x:xE Aor(r E B orx C)= E { x : x E AUA} = {x:x¬A orx EA} ={x:x¬A} =A = {x:xE (A U B) orx E C} RH.S. = {x:x¬ (A U B) U C} = (A U B) UC (i) LH.S. = AnA R.H.S. = {x:x¬ANA}= {x:x EA and x EA} ={x:xEA}=A (i) L.H.S. =AN(B nC) F RH.S {x:xEANBOC)}={x:x¬ Aand r E(B nC) IL ldentity Laws B) andx EC x:rEA and(r E Band x E C)} {:«EA and = r E Statement. If A is any set, then - {:xE (AN B) and x E C} () AUp=A (in) AnU=A Proof. (i) L.H.S. {x:x¬(A NB)nC} =(AN B) nc AUO= {x:xEAU¢} = R.H.S. = {x:x E A or xE p) = {x:x E A} V. Distributive Laws = A Statement. IfA, B, C are any three sets, then = R.H.S. (0AUBN)=(A U B) n (A UC) (i) L.H.S. = AnU= {x:x E ANU} (i) AN(B UC)=(A n B) U (A NC) ={x E A and x E U} = {x:xE A} Proof. L.H.S. =A U (B nC) rxEAU(BNC)} - {rixEAor x E (B n C)} = A = R.H.S. 17 16 SPECTRUM DISCRETE MATHEMAT. MAT AND FUNCTION - r:rE A or (1 and x C)}{*:(tEA or x E B) and (r E Aor xEC} E B E SETS, RELATION A = BNA that B - sets, then prove - {r:r¬ (AU B) and xE (A U C)} ir:x¬ (A U B) n (AU c)} = 1.9. IfA, B are two {(A UB) (A U C)} Proof. L.H.S. B -A = {r:xE B andr E A} £A} ¬Band x {x:x ¬ B-A)} {x:x = R.H.S. - AU(BOC) - (A U B) n (A UC) - {r:xE(B nA°)} Note. We can also prove above result by showing that - BA AU(B nC)c(AUB)n(A U C)and(A U B)N(AUC)CAU(B nnc) = R.H.S. (in L.H.S. =An(BUC) {x:xEAn(B UC)}=i*ixE Aandx E (B U C} B-A BNA that nc)=(A- B) U (A-C) x x E A and (r E B or x E C)} = {x:(*E A and x E B) or( r E A and x E C 1.10.1f A, B, C are any sets, pro (i) A-(B A-(BUC)=(A-B) n(A -C) - {x:x¬ (A B) or xE (AN C)} *:xE (AN= B) U(A N C)} () =(A B) U (A NC) Proof. () L.H.S. =A-(BUC) : B -A BOA R.H.S An(BUC) A BUC) = (A B) U(A NC) -AnB°nc) =(A nB) n(A NC') VI. De Morgan's Laws - ( A - B ) n (A - C) Statement. IfA and B are two sub-sets of U, then - R.H.S. () (AUB)" =A° nB° L.H.S. R.H.S. OR Complement of union of two sets is equal to the intersection of complements of two sets. A-(BUC)=(A -B) n (A-C) (i) (A n B)=A°UB° (i) L.H.S. =A-(Bnc)=A N(Bnc) OR -An(B°UC)=(A N B°) U(ANc') =(A - B) U(A -C) Complement of intersection of two sets is equal to the union of complements of two sets. R.H.S. Proof. () LH.S.= (A U B L.H.S. R.H.S. - {x:x¬ (AU B)} ={x:x £ (A U B)} = {x:x£A and x£ B} A-(B nC)=(A- B) U (A - C) {x:xEA andr E B°- {x:x¬ (A° OB)} AB ILLUSTRATIVE EXAMPLES = R.H.S. {1,5, 9} and let the universal set U = {0, 1, 2,. 9}. L.H.S. = R.H.S. Example 1. Let A ={0, 2, 3}, B = {2,3} and C = Determine (A U B) =A NB (a) AOB (b) AUB (c) BUA () AUC (e) AnC (ii) L.H.S. =(AnB ) A-B (8) B-A (h) A )C = tx:x E(A OB} {x:r £ (AN B)} ={x:x £Aorx £ B} = Sol.(a) ANB = { x : x E A and x E B} = {2,3} = :rEA'orxEB°} {r:xE(A°UB)} = 6) AUB {x:x E A or xE B} = {0,2, 3} = A° UB (c) BUA {r:x ¬B or x E A} = {0,2,3} = R.H.S. (d)AUC = {x:x¬A or x E C} = {0,2,3, 5, 9} L.H.S. = R.H.S. (e) AnC ={x: rE A and x E C} = {} orp (A B=A° UB 18 SPECTRUM DISCRETE MATHEM 19 ( A-B {rx ¬ {0}Aand x £ B} - RELATION AND FUNCTION SETS, (g) B-A {r:xEB and x£ A} ={) or o A NA Further letx be any element of (h) A = {r:xEU and x A} = {1,4, 5, 6, 7, 8,9 E A. x E¢ xE A and x rE A and x E A C x : r EU and x £A} = {0,2 ,3 ,4,6,7, 8} AACo Example 2. Let A = {r:r is an even integer) and let But o C ANA' always B {r:r is an integer divisible by 6 }. Let C= {x: X is an integer divisible by 2 or 3), and let D ANA=0. = U (B\A)U(ANB). an integer divisible by 2 and 3}. Determine which of the (AB) following relations hold. If containment A and B, prove that AUB= determine whether it is proper ent ho Example 4. Forsets A U (A N B) Sol. R.H.S. =(AB) U (B\A) (a) ACB (6) BCC(¢)CCB ()DCB («)ACD ()DCC (g) C C D. A) U(ANB)] =(A-B) U[(B- [A-B AOB] Sol. Here A = {r:r is an integer) ..-3,-2,1,0, 1,2,3,.. AIU(AN B)J =(A N B)U [(BN Commutative Law) Ber is an integerdivisible by6} f. - 12,-6,0,6, 12,.. =(AN B) U [(BN A)U (BNA)] (Distributive Law] C xx is aninteger divisible by 2 or 3} =(ANB)U [B n (A°UA)] [A UA X] 6,4,-3,-2,0,2, 3,4,6,. = (A NB)U (BNX) D={rris an integer divisible by 2 and 3} (ANB)UB Distributive Law] 12-6,0, 6, 12.. (A U B) n (B°U B) (a) ACB doesnotholdas -3EA but -3 £B =(A U B) nx (6) BCCholds as any integer which is divisible by 6 is also divisible by 2 or 3. = AUB (c) CCBdoes not hold as 2¬C but 2 ¢B. L.H.S (d) DCBholds since D =B . Let A and B be two sets. Prove that Example AAB ( A B) U (B - A). (e) ACDdoes not hold as 1EA but1 ¢D )DCCholds as any integer which is divisible by 2 and 3 is also divisible by Sol. A AB is symmetric difference of sets A and B. It is defined as the set of elements that belong to set A 2 or 3. or set B but not to both. (g) CCDdoes not hold as 2EC but 2 ¢ D. AAB {:(«E A and x £ B) or (r E B and x A)} Example 3. Prove that A UA' =U and A nA'= Sol. Let x - {x:(E A B)or(xEB- A)} be any element AUA' x: (xE A B) U (r EB - A)} x¬A or xE A xEA or xfA * x¬U = (A B) U (B - A) x EAUA x¬UJ Hence proved. AUASU .Example 6. For sets A, B and C using properties ofsets, prove that Conversely, let x be any element ofU ( A (BUC)=(A-B)n(A-C) i) A (BnC)=(A-B) U (A-C) either xEA or x£A * either E A (i) (A U B) - C=(A C)U(B - C). (iv) A-B= A - (A N B) x or rE A xEAUAA )AUB-(A -B) U B. . xEU xE AUA Sol. () A (B UC) =An (B UCy :X-Y=XnY] UCAUA An(B' nc) :(B UC) =B' nC] ...4) From (1) and (2) we get = (A N B') n(ANC) AUA=U - (A-B) n(A - C) 20 SPECTRUM DISCRETE MATHEM 21 (i)A -(Bn) =A n (BnCy FUNCTION A-Y = Xn SETS, RELATION AND that A = B An (B' UC) (BnCy = B' U (i) Assume UA =A = (A N B) U(A NC) : is distribution Over AUB A AOB ANA=A - (A-B) U (A-C) (AUB)-C =(A U B) nC AUB ANB () [: X- Y- Xn AB AUB ANB A UB =(A NC)U (BnC) two sets. Prove that (A - B) U B = Let A and B are any (A-C)U(B-C). Example8. NB)UB Sol. L.H.S. - (A -B) U B=(A (By Distributive Law) () A-B-A-(ANB) (A U B) N (B'UB) Let xEA -B (A U B) U=(A UB) iff rE A and x E B U B)-(C-A) = (A U B) N (C' U A). Example 9. Let A, B and C be subsets of Set U. Show that (A iff rE A and x E AOB Sol. L.H.S. = (A U B)- (C- A) = (A U B) -(CN A) iff rE A (AB) - (A U B) n (COA) = (A U B) n (c'u(A) (By De Morgan's Law) A-B A-(A NB) : ( A ) = A] () A UB (A -B) UB (A U B) n (C'UA) U B) (C A) =A U (B-C). A, B and C, show that (A - - Let xEAUB Example 10. For set B)- (C-A) = (A U B) - (Cn A') = (A U B)n (CnAY iff rE A or xEB Sol. L.H.S. =(A U Law iff xE A and xB or xEB (By De Morgan's - (A U B) n (C'u(A)) : A ' = A] iff rE A -B or x¬B = (A U B) n(C'UA) iff xE (A-B) UB (Commutative Law) = (A U B) n(AUC) AUB (A -B) U B (Distributive Law) =AU(Bnc) Example 7. Prove that AUB= ANBiff A =B = AU(B-C) Sol. () Assume that A UB= ANB = R.H.S. Let x be any element of A Example 11. Let A and B be the following subsets of the real numbers EA xEAUB A= {x:0<x<5} and B= {x:2 <x <8}. Express A U B as the union of three disjoint sets. xEAOB [ o f(l) Sol. A= {x:0<«5},B={tx:2<x<8} xEB We know that EA r¬B AUB-(AB) U (B\A) U (A NB) .ACB Here (A B) ={x:0<x s 2 . . Similarly B CA () (BA) = {x 5 Sx< 8} From (2) and (3), A = B. and A nB= {x:0<x<2} AUB ANB A B AUB= {x:0<x s 2} U {x: 5 sr<8} U fr: 0<x <2). 22 SPECTRUM DISCRETE MATHEMA 23 NT AND FUNCTION Example 12. Let X = {1, 2,3, 4) SETS, RELATION (BnC). OB)NCand A n If R=ia, PlrEXAyEX^(r-y) is an integral non-zero multiple of 2} Example 16. Draw venn diagram of (A (An B) n c S={x, y>xE X A y EX A (r-y) isan integral non-zero multiple of3} Sol. AnB Find R US and R nS. Sol. X= {1,2,3,4) R = , l x ¬X Ay EXnT-y) is an integral non-zero multiple of 2} ={1,3>,<2,4>} S {, y>|x EX AyEXn-y)is an integralnon-zero multiple of3} c = {<1,4>} An(BnC) BOC RUS ={<1,3>,<2, 4>, <1,4> and Rns={ } or o. Example 13. For A = {1,2, {1,3},p}, determine the following sets () A-{1} () A-o Gin) A-{#} () A-1,2} Sol. We know A-B {x:x¬A and x£B} ( A-1)-{2.{1,31.o} (in A-p =A (ii) A- (o}= {1,2, {1,3}} (h) A-1,2} {{1,3}.p} Example 17. If A, B are two sets, then show that AUB=¢ A=p, B=¢. Example 14. A= {1,2,3), B- {3,4, 5), U =- {1,2,3,4, 5, 6,7, 8,9. Sol. Let A UB=¢ Find A UB, AnB, A - B, A. We know A=p, B=¢ Sol AUB 1,2,3} U (3,4,5} {1,2,3,4,5} Let xE A AOB {1,2,3} n {3, 4, 5} = {3} rEAUB : ACAUB] A-B {1, 2.3 -{3,4, 5} = {1,2} [A UB ] A =U-A AS {1,2,3,4, 5, 6,7, 8,9- {1,2,3 Also SA 14,5,6,7,8,9) A Example 15. Find A U (BA) = A UB. Similarly, we can show B=° Sol. L.H.S. =AU(BA)= AU(B- A) AUB ¢ A=¢, B=¢ ...(1) AU(B A) Again, let A =p, B =¢ [A -B ANB (A U B)n(AUA) We show AUB [Distributive Law (A U B) nx Let xEAUB AUA= AUB ANX=A xEA orx¬B » r E¢ or x ¬p x Ep = R.H.S. AUBSO SPECTRUM DISCRETE MATHEMA 25 Also AUB SETS, RELATION AND FUNCTION {1,2,3,4. set P(A) of A = AUB Example 21. Find power (1,2,4). (2, 3, 4}. (1}. (2), (3}, (4), {1,2}. {1, 3}, {1,4}, 2, 3}, (2, 4},(3, 4}, {1,2,3}. A- . B- AUB=¢ Sol. PA)- {ø. 1,3,4).{1, 2, 3, 4} From (1) and (2), we have that AUB p A=p., B =o. B be two sets. Prove Example 22. Let A and Example 18. Let A { 1 , 2 . 4 } , B = (4, 5, 6}. Find A U B , A NBand A B. A = {1,2,3,4 A-B A n B Sol. Letx EA - B Sol. A = {,2.4} and B = {4,5,6} r¬AOB ¬A and x E B° AUB (1.2,4) U 14,5,6} = {1,2,4,5,6} r EA and r £B x AOB {1. 2 , 4} 14,5 , 6} = {4} A BSANB (i) Conversely let x E ANB, then A B {1.2,4} (4,5.6} = {1,2). (ui) x¬A and r£B ° xEA- B rEAand x ¬B° Example 19. Is it true that power set of A U B is equal to union of power sets of A and B ? Justify ANBSA - B ...(i) Sol. Let A = {a. b}, B = {c) A B ANB. then A U B-{a. b, c} 6 elements respectively, what can be the minimum Example 23. If A and B be two sets containing 3 and number of elements in AUB? Find also, the maximum number of elements in A U B. P(A)= {0. {a}. {b}. {a, b}} Sol. We haven (A U B) = n(A) + n(B) - MA N B). P B) 1o.C}} This shows that B) is minimum maximum according n(AN B) is maximum or minimum n(A U or as P(A)UP(B)= {o. ia}. {b}, fc). ta, b}} respectively. where as Case I: When n (A n B) is minimum, i.e. n (A N B) = 0. This is possible only when A n B = . In P(AUB)= {p. ta}.{b}, fe}. fa. b}, {a, c}, {b, c}, {a, b, c}} thiscase, n(A U B) n{A) +n(B)= - 0 n(A) + n{B) = 3 +6= 9. So, maximum number of elements in showing that P(AUB) = P (A) U P (B) AUBis 9. Example 20. Prove that P (A n B) = PA) N P B ) Case II: When n(A N B) is maximum. This is possible only when ACB. In this case, n(A n B) =3. Sol. Let X E P(ANB) mA U B)-mA) + n[B) - {ANB)=3+6-3 -6 then XC (AN B) so, minimum number of elements in AUB is 6. XCAand also X C B ( ANBCA and also ANBC xE MA) and also XE P(B) X EP(A)OPB) EXERCISE 1.2 1. Let A {0, 2,3}, B {2,3} and C= {1,5,9}. Let D {3, 2} and let = = Hence P(ANB) C P (A) n P (B) E {2,3, 2}. Determine which Conversely, let Y E P(A) O P(B) of the following are true. Give reason for your decisions. ()A=B (6) B =C () B=D ()B E (e)ANB=BNA YEPA) and YE P(B) YCA and YS B )AUB=BUA (g) A-B=B-A each element of Yis contained in both A & B each element of Yis contained in A NB 2. Determine whether each of the following inclusions is proper: YCANB YE P(ANB) () ACB where A {x:x is odd an prime} and B {r:x is an integer not divisible by 2} Hence P(A)N P(B) E P(ANB) (6) SCT, where S {x:r is a real number with a finite decimal expansion), From (1) and (2); we get and T= {x:x is a rational number, P(ANB) = P(A) N PB) (c)XCY, where X = {r:r is integer divisible by 9} and Y {x:x is an integer divisible by 3} = 27 SPECTRUM DISCRETE MATHEM FUNCTION 26 relations holds SETS, RELATION AND 3. of the following Prove that each ANSWERRS and B= {x:ris an integer multiple ot. (g) False {x:r is an integer multiple of 10} True (a)ACB where A True (e) True ) even integer} (c) True (d) integer) and B {x:r is an (b) False = False B, where A {r:ris even I. () (c)Notproper = (6) A (6) Not proper 7. 8,9). A {r EU: r multiple of 3}, = 2. (a) Proper c)0, 1,2} Let U= {0. 1.2.3.4 5.6. (b) 13,6,9 4 {0, 3, 4, 5, 6, 7, 8, 9} 4. (a) kl, o,p, q..X} B E U : r-5 0} {a, b,c , . j , 2 6. (a) th, i,j, k, } (b) Determine (a) A UB (b) AN B (c) B ( ) h , i, j, k,1, o, p, q, } (c){n, ij, k, }} P, 2} d, e,f, g, m, n, o, . . , defined as follows ){a, b, c, 5. Let A and B be subsets of natural numbers (e) {r, s, t , . } ifr is divisible by p, then x is divisible by p'} (h) km,n, 0,p, q} A r ifp is prime and (e) la, b, c, d, e, g} there is an integer ysuch that x =y'}. a , b, c d e.fg} )fa, b, C,. and B {x: containment is proper. 7. (a) {..12,-6, 0, 6, 12,.. Prove that BCA. Show that the Let A = {a, b, C,. },B = {h, i.j., q} (6) . - 9 , - 6 , -5,-3, 0, 3, 5, 6,. 6. Let U be the set of letters of the alphabet. Find the elements in each ofthe following set: (c).-15,-12, -6,0, 6, 12, 15,. and C = {o.P, q,. (d) - 12, 10, 6, -5, 0, 5, 6, 10, 12, - () (AnB) UC (e) A'OB (6) A UC (c) AnBUC) -11,-7, -5, - 1, 1,5, 7, 11,... (a) ANB (e) AB) (g) AB (h) BA ( A\(BIC) () AVCB) ){-7,-5,-3,-2,-1, 1,2, 3, 4, S,. and let A = {r:r is divisible by 3}, let (g) ....15, -9, -3, 3,9, 15,.. 7. Let U be the set ofintegers B : r is divisible by 21. Let C = {x: r is divisible by 5} Find the elements in each oft (h)..- 10, -8, -4,-2, 2, 4, 8, 10.. followingset: (9 - 1 5 , -9, -3,3,9, 15,..} (a) A NB (6) A UC (c) An(BUC) (d) (ANB) UC (e) A°NB ) f . . . - 18, -12,-9, -6,-3, 0,3, 9, 12, 18....} h) BA ) A(BIC) ) AVCB) 8. () False (6) True ( )True (d) False (e) False (AN B (g) AB (e) True (N True 9. (a) True (b) True (c) False () False 8. Answer true or false (a) AUB=(A UB 6b) A=UA AUBU)=(AUB) UC 1.11. Some Important Problems (e) AVBIC)-(A\B)\C ()AUBnC)=(A UB)nc We give below some other important problems on union and intersection. 9. Let A={a, b, c, d, e}, B {a. b}, C= {B.¢}, D = {a. b, {a, b}}. Find A n B, COD, A N D, CNP and D O P(A). Indicate whether, each ofthe following is true or false: ILLUSTRATIVE EXAMPLES (a) A EP (A) (b) CCP(A) (c)DCP(A) (d)B CD Example 1. Give examples ofthree sets A, B, C for which A -(B-C) = (A - B)-C (e) BED la, b} EC Sol. TakeA {1, 2, 3), B {3, 4, 5), C (6,7 10. Prove that if A CB and B C C, then A C C. B-C (3,4, 5} - {6, 7} {3,4, 5} 11. Prove that AB and BlA are disjoints. A-(B-C){1,2,3)-(3,4,5} {1,2} .(1) 12. Prove that if A C B, then P (A) c P (B). Also A-B ={1,2,31 13, 4, 5} ={1,2 13. Let A and B sets, then (A B) U(ANB)=A. ( A - B ) - C - { 1 , 2 ) - ( 6 , 7 } = {1,2} ...(2) From (1) and (2), 14. Let A, B, C be sets. If AS B and B nc =g, then AnC=¢. we get, Prove that A' - B' = B - A. A-(B-C)= (A - B ) -C 15. 28 SPECTRUM DISCRETE MATHEM EMATY 29 Eumpe 2. Give an example of three sets A. B and C such that SETS, RELATION AND FUNCTION AnB .BnC#p. AnC^p but A N BnC =¢ be any element of B Sol. Let x Sol Let A{1.2,3}, B ={3,4,5}.C= (4, 5,2 xE A. rE A or AB {3 p. B nC= {4, 5) * p, cOA= {2}*¢ EA Case I. x But A n BnC=¢ xEANB :AnB AnC] Eample3. Prove that ACB« B CA' for all sets A, B. r E ANC Sol () Assume that A CB >xEC We are to prove that B CA But x is any element ofB Letr be an element of B BCC Case I1. x £A xEAUB : x¬B xEAUC AUB= AUC] xA xEC x¬A] rEA But ris any element ofB BCA BCC ACB B CA from both the cases, it is clear that () Assume that B° C A BCC .. (1) We are to prove that A C B. Similarly CCB .(2) Lety by any element of A. From (1)and (2), B=C yEA y¬A AUB A UC and An B ANC B C. y£B :off Example 5. For any sets A and B, prove that (A-B) U (B-A) = (A U B)-(A N B) yEB ACB Sol. R.H.S. = (A U B) (A N B) =(A U B) n (AN B)' A-B =ANB'] - BCA' ACB - (A U B)nA" UB') =[(A UB)NA']u[(AUB) OB'] Combining the results proved in () and (ii), we get, =[(A NA')U(BNA')]u[(anB')U(BOB')} ACB B CA -[oU(Bn A' )]u[(A n B')UO] Example4. If A, B and C are any sets, prove that AUB A UC and AOB= ANC (BA')U(ANB)=(AOB)UBNA')=(A-B)U(B-A) BC L.H.S. SPECTRUM DISCRETE MATHE 31 30 AEMAN RELATION AND FUNCTION that AN(B-C) = (A N B)-(A NC) SETS, Eumple 6. Show R.H.S. (A OB)-(A nC) =(A NB)n(ANC=(AN B)n (A° uc) SECTION-II Sol. - [(A B)n A"]u[(AN B)nc]=[A NA')OB] u[A n (B nc RELATIONS (p n B)U [A N(B-C)]=¢U[An(B-)] 1.12. Ordered Pair An(B-C) denotes an ordered pair A and bEB, then (a, b) - L.H.S. and B be two given non-empty sets. If a E Let A is b. whose first component is a and the second component EXERCISE1.3 The ordered pair (a, b) and (b, a) are different unless a =b. Also two ordered pairs (a, b) and (c, d) if a = c and b=d. identities: are only equal if and 1 Verify the following whereas the sets {5, 2} and {2, 5} are equal. AUBnC)=(A UB) N (A UC) () AnBUC)=(A N B) U (A NC) Note 1. Ordered pairs (2, 5) and (5, 2) are not equal also and b may not be distinct i.e., (a, a) and (b, b) are where, A. B, C are three sets defined by Note 2. In the definition of an ordered pair, a ordered pairs. 7 {1.2,4,5 }, B {2,3,5, 6 }, C={4,5, 6, = A 1.13. Cartesian Product of Sets 2. are two sets, then find X n (XU Y). If X and Y Cartesian Product of Two Sets. If A and B are two non-empty sets, then the set of all ordered pairs (a, b) and is denoted by A x B.. Showthat )ACAUB (i) ANBCA. 3. where a E A and b E B is called the cartesian product of sets A and B 4. Prove the following: In symbols, A xB= {(a b): a E A, bE B} (i) ANBCB (i) BCAANB=B ( BCAUB Note 1. A x B and B x A are different sets if A * B. () C and B C D AC AUBC CUD (v)BCC>ANBCAnC 2. AxB= when one orboth of A, B are empty. (v) A =B ACB and BCA. Cartesian Product of Three Sets. The set of all ordered triplets (a, b, c), of elements 5. For any two sets A and B, prove that A nB=¢ * ACB. a EA, b E B, c E C is called the Cartesian product of the three sets A, B, C and is denoted by AXBxC. 6. Prove that in) A -(A - B) = A OB We know that ) An(A'UB) =ANB 7. Prove that A\ B' = B\A. A=A XA ={«,y):x,y E A} A=AxA XA={x, y, z):x, y, zE A} 8. Prove the following : (G) An(B- A) = ¢ ii) AU (B- A) = AUB 1.14. Graphical Representation of A xB ((A-B)nB=o Draw two perpendicular lines X'OX and Y'OY intersecting at (i) A-B =A - (A NB) ) (A-C)U(B-C)=(AUB)-C 0, where X'OX is horizontal and Y'OY is vertical. Now on --- (a, b) (v) (A- B)-C=A - (B U C) = (A - B) n A -C) (vii) IfA C B, then B -(B-A/" horizontal line X'OX represent the elements of A and on vertical 9. Prove that A n (B\C) (A N line Y'OY, represent the elements of B. = B)MAN C). Now if a E A, bB, draw a vertical line through a and a E ANSWERSs horizontal line through b. The point where they meet represents the ordered pair (a, b), The set of all such obtained 2. points graphically represents A x B.
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